Quantum mechanics is mathematically formulated in Hilbert space or projective Hilbert space.
The space of pure states of a quantum system is given by the one-dimensional subspaces of the corresponding Hilbert space (or the "points" of the projective Hilbert space).
In a two-dimensional Hilbert space this is simply the complex projective line, which is a geometrical sphere.

The Bloch sphere is a unit 2-sphere, with each pair of antipodal points corresponding to mutually orthogonal state vectors.
The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): |0\rangle
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): |1\rangle
, respectively,
which in turn might correspond e.g. to the spin-up and spin-down states of an electron.
This choice is arbitrary, however.
The points on the surface of the sphere correspond to the pure states of the system, whereas the interior points correspond to the mixed states.[2][3]
The Bloch sphere may be generalized to an n-level quantum system but then the visualization is less useful.

In optics, the Bloch sphere is also known as the Poincaré sphere and specifically represents different types of polarizations. See the Jones Vector for a detailed list of the 6 common polarization types and how they map on to the surface of this sphere.

Contents

Definition

Given an orthonormal basis, any pure stateFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): |\psi \rangle
of a two-level quantum system can be written as a superposition of the basis vectors
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): |0\rangle
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): |1\rangle
, where the coefficient or amount of each basis vector is a complex number.
Since only the relative phase between the coefficients of the two basis vectors has any physical meaning, we can take the coefficient of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): |0\rangle
to be real and non-negative.
We also know from quantum mechanics that the total probability of the system has to be one, so it must be that

Given this constraint, we can write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): |\psi \rangle
in the following representation:

on the unit sphere in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\mathbb {R}}^{{3}}
.

For mixed states, one needs to consider the density operator. Any two-dimensional density operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \rho
can be expanded using the identity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): I
and the Hermitian, tracelessPauli matricesFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\vec {\sigma }}
:

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\vec {a}}\in {\mathbb {R}}^{3}
is called the Bloch vector of the system. It is this vector that indicates the point within the sphere that corresponds to a given mixed state. The eigenvalues of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \rho
are given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\frac {1}{2}}\left(1\pm |{\vec {a}}|\right)
. As density operators must be positive-semidefinite, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): |{\vec {a}}|\leq 1
.

in accordance with the previous result.
Hence the surface of the Bloch sphere represents all the pure states of a two-dimensional quantum system, whereas the interior corresponds to all the mixed states.

A generalization for pure states

Consider an n-level quantum mechanical system. This system is described by an n-dimensional Hilbert spaceHn. The pure state space is by definition the set of 1-dimensional rays of Hn.

Theorem. Let U(n) be the Lie group of unitary matrices of size n. Then the pure state space of Hn can be identified with the compact coset space

To prove this fact, note that there is a naturalgroup action of U(n) on the set of states of Hn. This action is continuous and transitive on the pure states. For any state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): |\psi \rangle
, the isotropy group of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): |\psi \rangle
, (defined as the set of elements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): g
of U(n) such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): g|\psi \rangle =|\psi \rangle
) is isomorphic to the product group

In linear algebra terms, this can be justified as follows. Any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): g
of U(n) that leaves Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): |\psi \rangle
invariant must have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): |\psi \rangle
as an eigenvector. Since the corresponding eigenvalue must be a complex number of modulus 1, this gives the U(1) factor of the isotropy group. The other part of the isotropy group is parametrized by the unitary matrices on the orthogonal complement of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): |\psi \rangle
, which is isomorphic to U(n - 1). From this the assertion of the theorem follows from basic facts about transitive group actions of compact groups.

The important fact to note above is that the unitary group acts transitively on pure states.

Now the (real) dimension of U(n) is n2. This is easy to see since the exponential map

Let us apply this to consider the real dimension of an m qubit quantum register. The corresponding Hilbert space has dimension 2m.

Corollary. The real dimension of the pure state space of an m qubit quantum register is 2m+1 − 2.

The geometry of density operators

Formulations of quantum mechanics in terms of pure states are adequate for isolated systems; in general quantum mechanical systems need to be described in terms of density operators. However, while the Bloch sphere parametrizes not only pure states but mixed states for 2-level systems, for states of higher dimensions there is difficulty in extending this to mixed states. The topological description is complicated by the fact that the unitary group does not act transitively on density operators. The orbits moreover are extremely diverse as follows from the following observation:

Theorem. Suppose A is a density operator on an n level quantum mechanical system whose distinct eigenvalues are μ1, ..., μk with multiplicities n1, ...,nk. Then the group of
unitary operators V such that V A V* = A is isomorphic (as a Lie
group) to